Question

how to
prove n sigma r=1=n(n+1)/2

Answer 1

\[\sum_{r=1}^{n}r=n(n+1)/2\]

Answer 2

how to prove this

Answer 3

you may find this interesting :)
http://mathcentral.uregina.ca/QQ/database/QQ.02.06/jo1.html

Answer 4

but how to prove it

Answer 5

First use: \[
\sum r = 1+2+\dots +n = n + (n-1) + \dots +1 = \sum (n+1)-r
\]

Answer 6

Therefore \[
2\sum r = \sum r + \sum(n+1)-r = \sum r + (n+1) - r = \sum n+1 = n(n+1)
\]So \[
2\sum r = n(n+1) \implies \sum r = \frac{n(n+1)}{2}
\]